CONNECTED LIE GROUPS THAT ACT FREELY ON A PRODUCT OF LINEAR SPHERES.

J.P.C.GREENLEES AND J.A.PÉREZ

Abstract. In this note we prove that a compact connected Lie group G admits a free action on some product of linear spheres if and only if it is isomorphic to (𝕋^{k} × SU(2)^{l})/Z for some k and l and for some central elementary abelian 2subgroup Z with Z ∩ SU(2)^{l} = 1.

1. Introduction.

Let G be a compact Lie group. The complexity of G is reflected in the topological complexity of the spaces on which it acts freely. It is therefore valuable to have constructions of spaces E on which G acts freely, especially if E has simple topology. For example if E is contractible it is a universal free G-space, and classifies principal G-bundles, so that a convenient description of E allows calculation of the cohomology of the classifying space BG ≃ E/G. Milnor’s construction of the universal space is natural and is well suited to ordinary cohomology, however it can be convenient to have a ‘smaller’ model or one tailored for use with a complex orientable theory.

For this we use unit spheres S(V) of orthogonal representations V as building blocks. For instance the groups 𝕋 of unit complex numbers and Sp(1) ≅ SU(2) of unit quaternions each have a representation V so that they act freely on S(V), S(V ⊕ V), S(V ⊕ V ⊕ V), …, and hence on their union S(∞V), which is contractible. In this case the periodic cohomology of the classifying space BG is reflected in the construction. Groups which act freely on a single linear sphere are very rare: if G acts freely on S(V) so does any subgroup, so that since simple representations of abelian groups are 1-dimensional, every abelian subgroup is cyclic. To cover a wider class of groups we ask if there are representations V_{1}, V_{2}, …, V_{n} so that G acts freely on S(V_{1}) × S(V_{2}) × ⋯ × S(V_{n}); then S(∞V_{1}) × S(∞V_{2}) × ⋯ × S(∞V_{n}) is a model for the universal free G-space. It is not hard to see that if G is a finite p-group or a torus extension over a finite p-group that G acts freely on a product of linear spheres in this way, and this is enough to make the construction valuable in equivariant topology, since general problems can often be reduced to this case by transfer.

Nonetheless the problem of determining which groups act freely on a product of linear spheres is of some interest in general. If G is finite, the best result is due to U.Ray who proved [5] the conjecture of J.G.Thompson, that if G acts freely on a product of spheres then the only possible nonabelian composition factors in G are P SL_{2}(5) and P SL_{2}(9). Since SL_{2}(5) acts freely on a single linear sphere and SL_{2}(9) acts freely on a product of two linear spheres this result is best possible. Accordingly we say that G is a Ray group (or locally smooth in the terminology of [2]) if G acts freely on S(V_{1}) × S(V_{2}) × ⋯ × S(V_{n}) for some n and some orthogonal representations V_{1}, …, V_{n}.

1991 Mathematics Subject Classification. 22E46, 57S25.

In the present paper we consider the much easier case when G is compact and connected. The basic observation is the following.

Theorem 1.1. Let G be a compact connected Lie group. If G is a simple Ray group then G ≅ SU(2).

We prove this in Sections 3 and 4.

From this it it not hard to deduce a complete classification of compact connected Ray groups; this is done in Section 5.

Theorem 1.2. A compact connected Lie group is a Ray group if and only if it has the form (𝕋^{k} × SU(2)^{l})/Z for some natural numbers k and l and some central elementary abelian 2-subgroup Z with Z ∩ SU(2)^{l} = 1.

It is clear that a subgroup of a Ray group is a Ray group, so that much of Theorem 1.1 follows by finding finite subgroups of G which are known not to be Ray groups. For some purposes we want a stronger result; we say that G is an almost Ray group if G has representations V_{1}, …, V_{n} so that the isotropy groups for the action of G on S(V_{1}) × S(V_{2}) × ⋯ × S(V_{n}) are finite. Our proof of the above theorems yield a stronger result.

Theorem 1.3. If G is a compact connected group which is not a Ray group then G contains a subgroup isomorphic to SO(3) or to Spin(5).

Since SO(3) and Spin(5) are not even almost Ray groups (see Section 4), this implies that allowing almost free actions is no help for connected groups.

Theorem 1.4. Let G be a compact connected Lie group. If G is an almost Ray group it is also a Ray group.

Accordingly the only compact connected almost Ray groups are those listed in Theorem 1.2.

Because the subgroup of a Ray group is a Ray group it is interesting to identify non-Ray groups whose proper subgroups are all Ray groups. The groups SO(3) and Spin(5) contain finite non-Ray groups (2.4 and 6.3), so that no compact connected group is minimal amongst non-Ray groups; on the other hand SO(3) and Spin(5) are minimal amongst groups which are not almost Ray groups (6.1).

The rest of the paper is arranged as follows. We begin in Section 2 by reducing the topology to representation theory. In the short Section 3 we observe that the classification of compact connected Lie groups reduces all proofs to consideration of a few cases, and explain the strategy of all remaining proofs. In Section 4 we make some observations about SO(3) and Spin(5) which complete the proof of Theorem 1.1 and of the fact that Theorem 1.3 implies Theorem 1.4. Next we spend Section 5 in a study of compact connected groups with all simple factors SU(2), showing that either they contain a subgroup isomorphic to SO(3) or they have enough representations to be Ray groups. Finally in Section 6 we return to Spin(5), and answer some natural questions about it.

2. Preliminary observations.

We begin by removing all references to topology. It is clear that S(V)^{H} = S(V^{H}), so that

S(V)^{H} ≠ ∅ ⇔ V^{H} ≠ 0.

Note too that if K ⊆ H then V^{H} ⊆ V^{K} so that it is only minimal groups K that need attention. Finally since (V ⊕ W)^{K} = V^{K} ⊕ W^{K} we may concentrate on simple representations. We may thus translate the definitions into representation theory.

Lemma 2.1. (i) A group G is a Ray group if and only if for every subgroup K of prime order there is a simple representation V of G with V^{K} = 0. (ii) A group G is an almost Ray group if and only if for every subgroup K ≅ 𝕋 there is a simple representation V of G with V^{K} = 0.

We say that a representation V of G detects a subgroup K if V ≠ 0 and V^{K} = 0. Proof: We prove Part (i); Part (ii) is simpler.

If G acts freely on S(V_{1}) × S(V_{2}) × ⋯ × S(V_{n}) then for each subgroup K we find S(V_{i})^{K} = ∅ for some i, and if V′_{i} is a simple summand of V_{i} then (V′_{i})^{K} = 0.

For the converse the main point is to establish that all subgroups K can be detected using finitely many representations. Note that translation by g gives a linear isomorphism V^{g^{−1}Kg} ⟶^{≅} V^{K} so that it is sufficient to find a representation for each conjugacy class and verify it detects a representative K of that conjugacy class. Now note that any subgroup of prime order is conjugate to a subgroup of a chosen maximal torus T. If G is of rank r so that T ≅ 𝕋 × ⋯ × 𝕋, let K_{1}, …, K_{r} be subgroups of order 2 with K_{i} non-trivial only in the i-th factor. Now choose simple representations V_{i} so that V_{i}^{K_{i}} = 0. It follows that the kernel L of the homomorphism T ⟶ GL_{1}(ℂ)^{r} defined by the action of T on the representations V_{i} is finite: otherwise there would be a circle in L which would contain an element of order 2. Choose simple representations V_{r+1}, …, V_{n} so that for each subgroup K of L with prime order V_{i}^{K} = 0 for some i. It follows that for any nontrivial subgroup H of G we have V_{i}^{H} = 0 for some i. Replacing V_{i} by an orthogonal representation by averaging we see that G acts freely on S(V_{1}) × S(V_{2}) × ⋯ × S(V_{n}).

Using Lemma 2.1 it is easy to check from a character table whether or not a given group is a Ray group. We shall need a source of non-Ray groups, and it seems worth proving an easy observation whose usefulness the first author learnt from R.A. Parker.

Lemma 2.2. If V is a faithful representation of G and ω is an involution with V^{ω} = 0 then ω is central.

Proof: Since V^{ω} = 0, the element ω acts as −1 on V by Schur’s lemma, and so commutes with the action of all other group elements.

Corollary 2.3. If G is a Ray group and ω ∈ G is an involution then ω is non-trivial and central in some quotient of G.

This, together with a lemma of B.Hartley, and a theorem of Griess is the starting point of Ray’s analysis. However we shall be satisfied with an immediate consequence.

Example 2.4. The alternating group A_{4} is not a Ray group. By the Feit-Thompson Odd Order Theorem, no non-abelian simple group is a Ray group.

3. The strategy of proof

In this section we use the classification of compact connected Lie groups, to reduces us to the consideration of a small collection of minimal groups, and summarize the work still to be done.

Lemma 3.1. Any compact connected simple Lie group G ≅ S̸U(2) contains a subgroup isomorphic to SO(3) or Spin(5).

Proof: Indeed, the only simple groups of rank 1 are SU(2) and P SU(2) ≅ SO(3), whilst any compact connected simple Lie group of rank 2 or more contains a subgroup of local type A_{2} or B_{2} (since G_{2} contains SU(3)), and hence a subgroup SU(3), P SU(3), Spin(5) or P Spin(5) ≅ SO(5). Viewing a real matrix as a complex one gives an injective homomorphism SO(3) ⟶ SU(3), so that both SU(3) and P SU(3) contain copies of SO(3).

Evidently the above proof only used the facts that any simple group of rank at least 2 contains one of rank 2, and a little bit about the rank 2 groups. Much of this can be deduced by considering centralizers of singular elements of a group, and this point of view highlights the representation theoretic idea. We have used the classification so as to be completely specific in the subsequent discussion.

Now we may outline the proofs of the theorems.

By Lemma 3.1, Theorem 1.1 will follow once we show that neither SO(3) nor Spin(5) are Ray groups. In fact we shall show in the next section that they are not even almost Ray groups, so that in addition Theorem 1.4 follows from Theorem 1.3.

To establish Theorem 1.3, recall that if G is any compact connected Lie group it contains a subgroup locally isomorphic to any of its simple factors. Consider the simple factors in a non-Ray group: if there is a simple factor S other than SU(2) then by 3.1 the group G contains a quotient of SO(3) or Spin(5), and the conclusion of 1.3 holds since the only proper quotient of these groups is SO(5). The other alternative is that all simple factors are isomorphic to SU(2): the analysis of Section 5, shows that any group of this form is either a Ray group or contains SO(3). Section 5 identifies these alternatives as stated in Theorem 1.2.

4. Representations of the groups SO(3) and Spin(5)

In this section we examine the representation theory of SO(3) and Spin(5), thereby completing the proofs of Theorem 1.1 and the fact that 1.3 implies 1.4.

The group G = SO(3).

Since SO(3) contains A_{5}, it is not a Ray group, but it is not much harder to see it is not even an almost Ray group. Indeed it is of rank 1, so that it has only one conjugacy class of circle subgroups, and it is enough to observe that the maximal torus fixes a vector in any simple representation V.

This is easily verified using the usual apparatus of representation theory, which we should briefly recall to fix notation. Let G be a compact connected Lie group and choose a maximal torus T; any representation V is determined by its restriction V |_{T}, and the representations of T which occur are called the weights of V. Representations of T may be identified with homomorphisms T ⟶ S^{1}, and, by passage to Lie algebras, with elements of LT^{∗}, where LT is the Lie algebra of T. In particular, the non-zero weights of the adjoint representation of G on its Lie algebra LG are called the roots of G and we let R denote the set of these. One may choose a linearly independent subset S ⊆ R ⊆ LT^{∗} of simple roots so that every root is of the form Σ_{σ∈S}m_{σ}σ where the m_{σ} are integers with the same sign; the set of positive roots R_{+} is then the set of roots for which the non-zero m_{σ} are all positive.

The case G = SO(3) is very straightforward; SO(3) has two roots α and −α, and we choose one of them (α say) as the simple root. G has a simple representation V of each odd complex dimension, and that of dimension 2n + 1 has weights −nα, (−n + 1)α, …, −α, 0, α, …, nα. Since 0 is a weight of any simple representation V, the maximal torus T fixes a vector in V. This should be contrasted with the case G = SU(2) which has in addition one simple representation V of each even dimension 2n, which has weights (−n + 1/2)α, (−n + 3/2)α, …, (−1/2)α, (1/2)α, …, (n − 1/2)α, and hence V^{T} = 0.

Note also that this result is best possible since the isotropy groups for the action of SO(3) on the unit sphere of the canonical representation on ℝ^{3} are all circles.

The group G = Spin(5).

Now consider G = Spin(5). We may choose coordinates for LT^{∗} ≅ ℝ^{2} so that its simple roots are α = (−1, 1) and β = (1, 0), and the set of all roots is R = {(x, y) | x, y ∈ {1, 0, −1}} \ {(0, 0)}.

[figure: α; β; α′]

Lemma 4.1. The circle S whose Lie algebra is ker(α : LT ⟶ ℝ) fixes a nonzero vector in every representation of G.

Proof: If V is a simple representation of G we consider its restriction V |_{Spin(4)} to the subgroup Spin(4) ≅ SU(2) × SU(2) of the same rank whose four roots are ±α and ±α′ where α′ = α + 2β. From the represention theory of SU(2) we know that if mα + nα′ is a weight of V where m and n are integers, then 0 is also a weight. The possible weights of a representation V of Spin(5) are the sums (m/2)α + (n/2)α′ for integers m and n. Using the representation theory of SU(2) × SU(2) again we see that each representation has a weight 0, α/2, α′/2 or β = (α′ − α)/2. Since the set of weights of V is invariant under the Weyl group, if α′/2 is a weight so too is α/2. Finally, since β is a root of Spin(5), if β is a weight of V so is 0 (see for instance [4, Proposition 21.3]). In short, either 0 is a weight of V or α/2 is a weight of V, and in either case the circle S with Lie algebra ker(α/2 : LT ⟶ ℝ) fixes non-zero vectors in V.

5. Classification of connected Ray groups

In this section show that any compact Lie group with all simple factors SU(2) either contains a subgroup isomorphic to SO(3) or has enough representations to ensure it is a Ray group: this completes the proof of Theorem 1.2.

Let G be a compact connected Ray group. By the structure theorem for compact connected Lie groups, G = G̃/Z where G̃ is of form 𝕋^{k} × H for a simply connected semisimple group H and Z is a finite central subgroup of G̃. In particular, if S is a simple factor of H then G contains a subgroup locally isomorphic to S. By Theorem 1.1 it follows that H ≅ SU(2)^{l} for some l.

Now since SU(2) has centre C_{2}, it follows that Z ⊆ 𝕋^{k}×(C_{2})^{l}. Since any quotient of a torus is a torus we may assume, by passing to quotient if necessary, that Z ∩𝕋^{k} = 1. Next we may assume that Z is a 2-group; indeed, any element of odd order in 𝕋^{k}×(C_{2})^{l} must lie in 𝕋^{k}, whilst the set of odd order elements of Z is a subgroup since Z is abelian. Finally observe that every element z of Z is of order 2, since z^{2} ∈ 𝕋^{k}. Thus we may suppose that Z is a subgroup of the elementary abelian 2-group E × F with Z ∩ E = 0, where E ≅ (𝔽_{2})^{k} is the set of elements of order ≤ 2 in 𝕋^{k} and F ≅ (𝔽_{2})^{l} is the set of elements of order ≤ 2 in SU(2)^{l}.

Lemma 5.1. Since G is a Ray group

Z ∩ F = 0.

Proof: Indeed if 0 ≠ z ∈ Z ∩ F then we may choose an order of the factors so that z = (1^{k}, −1^{a}, 1^{l−a}) for some a > 0. But then the image of the diagonal subgroup

SU(2) ⊆ SU(2)^{a} ⊆ 𝕋^{k} × SU(2)^{a} × SU(2)^{l−a}

in G̃/⟨z⟩ (and hence also in G) is SO(3), contradicting the fact that G is a Ray group.

One may try to put Z into some sort of standard form using automorphisms from GL_{k}(ℤ) on the torus factor and permutations on the SU(2)^{l} factor, but it does not seem easy to give a canonical form. For instance when k = 1 and Z is of order 2 we obtain the groups (𝕋 × SU(2)^{a})/⟨(−1, −1^{a})⟩ × SU(2)^{l−a} for a = 0, 1, …, l, no two of which are isomorphic.

Finally we show that the remaining possibilities are Ray groups.

Proposition 5.2. For any subspace Z ⊆ E × F with Z ∩ E = 0 and Z ∩ F = 0 the group G = G̃/Z is a Ray group.

Proof: We must find enough representations to detect all subgroups of prime order in G.

The simple representations of G are the simple representations of G̃ which contain Z in their kernel, and the simple representations of G̃ are tensor products of those of the factors. We let μ(n_{1}, …, n_{k}; a_{1}, …, a_{l}) denote the representation p(n_{1}) ⊗ ⋯ ⊗ p(n_{k}) ⊗ λ(a_{1}) ⊗ ⋯ ⊗ λ(a_{l}) where p(n) is the 1-dimensional representation of 𝕋 in which z ∈ 𝕋 acts as z^{n} and λ(a) is the (a + 1)-dimensional simple representation of SU(2). Note that −1 acts as (−1)^{n} on p(n) and as (−1)^{a} on λ(a), so that if z = ((−1)^{τ_{1}}, …, (−1)^{τ_{k}}; (−1)^{σ_{1}}, …, (−1)^{σ_{l}}) is of order 2, the condition that z act trivially on μ(n_{1}, …, n_{k}; a_{1}, …, a_{l}) is the mod 2 linear condition n_{1}τ_{1} + ⋯ n_{k}τ_{k} + a_{1}σ_{1} + ⋯ a_{l}σ_{l} = 0. The condition that μ(n_{1}, …, n_{k}; a_{1}, …, a_{l}) contains Z in its kernel is thus the requirement that the mod 2 reduction of (n_{1}, …, n_{k}; a_{1}, …, a_{l}) lies in the orthogonal complement of Z.

Now suppose x ∈ G is of prime order p, and that we want to detect the subgroup K = ⟨x⟩. Choose x̃ ∈ G̃ lying over x. Then x̃ is of order p or 2p, and if p is odd we can replace x by x^{2} without changing K and hence suppose x̃ is of order p. It is thus sufficient to detect elements of prime order in G̃ not in Z, and elements of order 4 whose squares are in Z. Consider x̃ = (t; s) = (t_{1}, …, t_{k}; s_{1}, …, s_{l}). If t is not of order 1 or 2 we may choose an order of the factors so that t_{1} ≠ ±1, and the element is detected by μ(2, 0, … 0; 0, …, 0); this accounts for most elements of odd prime order and all those of order 4 since Z ∩ F = 0. The other elements of odd prime order are of form (t; s) = (1; s), and we may choose an order of the factors so that s_{1} ≠ 1. The representations of form μ(n_{1}, …, n_{k}; 1, 0, …, 0) all detect (1; s), and their mod 2 reductions form an affine subspace of dimension k in E × F. Since Z ∩ F = 0 the subspace Z is of dimension < k and so its orthogonal complement is of dimension > n − k, so one of the specified detecting representations has Z in its kernel. This leaves the elements of order 2 not in Z, and since each of these has a representative ((−1)^{τ_{1}}, …, (−1)^{τ_{k}}; (−1)^{σ_{1}}, …, (−1)^{σ_{l}}) in the orthogonal complement of Z we may use μ(τ; σ) to detect it.

6. Further remarks on Spin(5).

In this section we resolve some remaining questions by showing (a) that Spin(5) is minimal amongst compact connected non-Ray groups, (b) that it is not minimal amongst all non-Ray groups and (c) that Spin(3) fixes a point for any action of Spin(5) on a product of linear spheres.

We continue to use the notation introduced in Section 4, but let us now be more specific, supposing that Spin(5) is constructed from the Clifford algebra for the negative definite form on ℝ^{5} with orthonormal basis e_{1}, e_{2}, e_{3}, e_{4}, e_{5} in the usual way. We take the maximal torus to be

T = {(cosθ − e_{1}e_{2}sinθ)(cosϕ − e_{3}e_{4}sinϕ) | 0 ≤ θ, ϕ ≤ 2π},

and then α corresponds to the form (θ, ϕ) ⟼ ϕ − θ whilst β corresponds to the form (θ, ϕ) ⟼ θ. Thus the circle S of 4.1 consists of points of T with θ = ϕ, it is singular, and its centralizer has identity component S^{1} × SU(2) ⊆ Spin(4). The other type of singular circle is represented by S^{∗} consisting of points of T with ϕ = 0; it has Lie algebra ker(α + β : LT ⟶ ℝ) and has centralizer whose identity component is the copy of U(2) which doubly covers SO(2) × SO(3) ⊆ SO(5).

Lemma 6.1. The group Spin(5) contains no copy of SO(3), and is therefore minimal amongst groups which are not almost Ray groups.

Proof: If there was a subgroup isomorphic to SO(3) then it would be a subgroup of a centralizer of some singular circle.

Next we construct a finite subgroup G̃ of Spin(5), and show that it is not a Ray group; we thank N.P.Strickland for suggesting G̃.

Let G denote the subgroup of monomial matrices in SO(5) (i.e. matrices with a single nonzero entry in each row or column, each of which is ±1). Let Δ denote the subgroup of G whose non-zero entries are all on the diagonal, so that Δ is elementary abelian of order 16; we thus have an extension

1 ⟶ Δ ⟶ G ⟶ Σ_{5} ⟶ 1.

Transpositions from Σ_{5} are represented in SO(5) by matrices

( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )

inserted in some pair of rows and columns.

Now let G̃ denote the inverse image of G in Spin(5), so that we have an extension

1 ⟶ Δ̃ ⟶ G̃ ⟶ Σ_{5} ⟶ 1.

Here Δ̃ is an extraspecial group of order 32 and its elements are ±1, ±e_{i}e_{j} and ±e_{i}e_{j}e_{k}e_{l} (where i, j, k and l are distinct).

Lemma 6.2. The elements of order 2 in Δ̃ are −1 (which is central) and the 4-fold products ±e_{i}e_{j}e_{k}e_{l} (all ten of which are conjugate).

Proof: The orders are easily verified. To see that e_{1}e_{2}e_{3}e_{4} is conjugate to ±e_{i}e_{j}e_{k}e_{l} for each set of indices we use any element over the appropriate permutation in Σ_{5}, and the e_{1}e_{2}e_{3}e_{5}-conjugate of e_{1}e_{2}e_{3}e_{4} is −e_{1}e_{2}e_{3}e_{4}.

Proposition 6.3. The group G̃ is not a Ray group.

Proof: In fact we show that there is no representation V ≠ 0 of G̃, so that V^{ω} = 0 where ω = e_{1}e_{2}e_{3}e_{4}. As remarked in 2.2, if V exists and N = ker(V) then the image \overline{ω} of ω is non-trivial and central in \overline{G} = G̃/N.

First we claim that N ∩ Δ̃ is either the centre Z = {1, −1} or trivial. Indeed, by 6.2 the subgroup N does not contain any element ±e_{i}e_{j}e_{k}e_{l}. Similarly if e_{1}e_{2} ∈ N then, by conjugating with an element over a suitable permutation, ±e_{3}e_{4} ∈ N, which contradicts the fact that ω ∉ N.

However N ∩ Δ̃ ≠ 1 since then the image of Δ̃ in \overline{G} would be isomorphic to Δ̃, and \overline{ω} would not be central in it. Thus N ∩ Δ̃ = Z, and we may as well work with the image \overline{N} of N in G. Because it is normal, the image \overline{\overline{N}} of \overline{N} in Σ_{5} is 1, A_{5} or Σ_{5}. We eliminate these possibilities in turn.

Firstly \overline{\overline{N}} ≠ 1 since \overline{ω} is not central in G, and \overline{\overline{N}} ≠ Σ_{5} since the number of −1’s in a monomial matrix has parity equal to the sign of the permutation it maps to: the quotient map G ⟶ Σ_{5} does not split. Finally if \overline{\overline{N}} = A_{5} then \overline{ω} is not central in G/\overline{N} since any transposition (such as (45)) represents the nontrivial element in Σ_{5}/A_{5}, and the conjugate of \overline{ω} by any element of G lying over (45) is the distinct element diag(−1, −1, −1, +1, −1).

Finally we consider the isotropy groups occurring in the action of Spin(5) on products of unit spheres. First observe that, if V is the 4-dimensional representation with highest weight α′/2, then the isotropy groups for the action of Spin(5) on S(V) must be of dimension 3 or more; since they are of rank 1 by 4.1, they must be locally isomorphic to SU(2).

Lemma 6.4. (i) If V is the 4-dimensional representation with highest weight α′/2 then the isotropy groups for the action of Spin(5) on S(V) are conjugates of Spin(3) and therefore S(V) = Spin(5)/Spin(3). (ii) If W is any representation of Spin(5) then Spin(3) fixes a point of S(W).

Proof: The fact that S(V) = Spin(5)/Spin(3) will follow from (ii). By dimension S(V) is a homogeneous space Spin(5)/H where H has identity component Spin(3), and since S(V) is simply connected H must be connected.

We now turn to the proof of (ii). To be specific about the subgroup, we mean

Spin(3) = Spin(3) × 1 ⊆ Spin(3) × Spin(3) ≅ Spin(4) ⊆ Spin(5).

Let ±γ be the roots of Spin(3). Thus Spin(3) is an isotropy group of S(W) if and only if the trivial representation is a summand of W |_{Spin(3)}, which happens if and only if the multiplicity of the weight 0 is greater than that of γ.

We remarked above that Spin(4) is a subgroup of rank 2 and its roots are ±α and ±α′. Thus if T_{1} denotes the maximal torus of Spin(3) the map p : (LT)^{∗} ⟶ (LT_{1})^{∗} induced by inclusion is projection parallel to ±α′. For a vector v ∈ (LT)^{∗}, let l_{v} denote the line through v parallel to α′. Thus Spin(3) is an isotropy group for the action on S(W) if and only if the sum of multiplicities of weights on the line l_{0} is greater than the sum of multiplicities of weights on l_{α}.

Using the subgroup 1 × Spin(3) it is clear that the sum of multiplicities on l_{0} is at least as big as that on l_{α}. It is easily verified in any particular case that it is actually greater, but for the general case we appeal to the explicit formulae of [1, p.1569]. In their notation, the various F (a, b) all contribute at least as much to l_{0} as to l_{α} whilst all representations involve a term F (0, b), which contributes one more to l_{0} than to l_{α}.

Corollary 6.5. The subgroup Spin(3) of Spin(5) fixes a point in any product of linear spheres.

References

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3. T. Bröcker & T. tom Dieck, Representations of Compact Lie Groups, Springer Verlag, GTM 98, 1985.

4. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer Verlag, GTM 9, 1980.

5. U. Ray, “Free linear actions of finite groups on products of spheres”, J. Algebra 47 (1992) 456-490.

Pure Mathematics, Hicks Building,

Sheffield. S3 7RH UK

e-mail: j.greenlees@sheffield.ac.uk

Universidad a. de Zacatecas, Escuela de matemáticas, Area de ingeniería no. 8, 98000 Zacatecas, Zac.

México.

e-mail: japerez@bufa.reduaz.mx